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Mirrors > Home > MPE Home > Th. List > 0cld | Structured version Visualization version GIF version |
Description: The empty set is closed. Part of Theorem 6.1(1) of [Munkres] p. 93. (Contributed by NM, 4-Oct-2006.) |
Ref | Expression |
---|---|
0cld | ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dif0 4286 | . . 3 ⊢ (∪ 𝐽 ∖ ∅) = ∪ 𝐽 | |
2 | 1 | topopn 21511 | . 2 ⊢ (𝐽 ∈ Top → (∪ 𝐽 ∖ ∅) ∈ 𝐽) |
3 | 0ss 4304 | . . 3 ⊢ ∅ ⊆ ∪ 𝐽 | |
4 | eqid 2798 | . . . 4 ⊢ ∪ 𝐽 = ∪ 𝐽 | |
5 | 4 | iscld2 21633 | . . 3 ⊢ ((𝐽 ∈ Top ∧ ∅ ⊆ ∪ 𝐽) → (∅ ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ ∅) ∈ 𝐽)) |
6 | 3, 5 | mpan2 690 | . 2 ⊢ (𝐽 ∈ Top → (∅ ∈ (Clsd‘𝐽) ↔ (∪ 𝐽 ∖ ∅) ∈ 𝐽)) |
7 | 2, 6 | mpbird 260 | 1 ⊢ (𝐽 ∈ Top → ∅ ∈ (Clsd‘𝐽)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4243 ∪ cuni 4800 ‘cfv 6324 Topctop 21498 Clsdccld 21621 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-top 21499 df-cld 21624 |
This theorem is referenced by: cls0 21685 indiscld 21696 iscldtop 21700 iccordt 21819 isconn2 22019 tgptsmscld 22756 mblfinlem2 35095 mblfinlem3 35096 ismblfin 35098 |
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